How to Construct Polar Codes
TL;DR: A method for efficiently constructing polar codes is presented and analyzed, proving that for any fixed ε > 0 and all sufficiently large code lengths n, polar codes whose rate is within ε of channel capacity can be constructed in time and space that are both linear in n.
Abstract: A method for efficiently constructing polar codes is presented and analyzed. Although polar codes are explicitly defined, straightforward construction is intractable since the resulting polar bit-channels have an output alphabet that grows exponentially with the code length. Thus, the core problem that needs to be solved is that of faithfully approximating a bit-channel with an intractably large alphabet by another channel having a manageable alphabet size. We devise two approximation methods which “sandwich” the original bit-channel between a degraded and an upgraded version thereof. Both approximations can be efficiently computed and turn out to be extremely close in practice. We also provide theoretical analysis of our construction algorithms, proving that for any fixed e > 0 and all sufficiently large code lengths n, polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n.
Citations
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03 Oct 2011
TL;DR: It appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes, and devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O( L · n) space.
Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, up to L decoding paths are considered concurrently at each decoding stage. Simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of L. Thus it appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the L “best” paths. In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, straightforward implementation still requires O(L · n2) time, which is in stark contrast with the O(n log n) complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O(L · n) space.
1,338 citations
TL;DR: Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of L, and it is shown that such a genie can be easily implemented using simple CRC precoding.
Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, $L$ decoding paths are considered concurrently at each decoding stage, where $L$ is an integer parameter. At the end of the decoding process, the most likely among the $L$ paths is selected as the single codeword at the decoder output. Simulations show that the resulting performance is very close to that of maximum-likelihood decoding, even for moderate values of $L$ . Alternatively, if a genie is allowed to pick the transmitted codeword from the list, the results are comparable with the performance of current state-of-the-art LDPC codes. We show that such a genie can be easily implemented using simple CRC precoding. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit, and then uses a pruning procedure to discard all but the $L$ most likely paths. However, straightforward implementation of this algorithm requires $\Omega (L n^{2})$ time, which is in stark contrast with the $O(n \log n)$ complexity of the original successive-cancellation decoder. In this paper, we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem: we devise an efficient, numerically stable, implementation of the proposed list decoder that takes only $O(L n \log n)$ time and $O(L n)$ space.
1,263 citations
TL;DR: It is shown that Gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them.
Abstract: Polar codes are shown to be instances of both generalized concatenated codes and multilevel codes. It is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes. Additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance. In some cases this also results in complexity reduction. It is shown that Gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them.
664 citations
Cites background from "How to Construct Polar Codes"
...An implementation of density evolution with complexity O(nμ(2) logμ) was proposed in [4], where n is the length of the polar code to be constructed, and μ is the number of quantization levels, which has to be selected sufficiently high to achieve the required accuracy....
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...It was shown in [4] that density evolution for polar codes can be implemented with complexity O(nμ(2) logμ), where μ is the number of quantization levels, which has to be set sufficiently high to avoid catastrophic loss of precision....
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TL;DR: This paper identifies and provides a detailed description of various potential emerging technologies for the fifth generation communications with SWIPT/WPT and provides some interesting research challenges and recommendations with the objective of stimulating future research in this emerging domain.
Abstract: Initial efforts on wireless power transfer (WPT) have concentrated toward long-distance transmission and high power applications. Nonetheless, the lower achievable transmission efficiency and potential health concerns arising due to high power applications, have caused limitations in their further developments. Due to tremendous energy consumption growth with ever-increasing connected devices, alternative wireless information and power transfer techniques have been important not only for theoretical research but also for the operational costs saving and for the sustainable growth of wireless communications. In this regard, radio frequency energy harvesting (RF-EH) for a wireless communications system presents a new paradigm that allows wireless nodes to recharge their batteries from the RF signals instead of fixed power grids and the traditional energy sources. In this approach, the RF energy is harvested from ambient electromagnetic sources or from the sources that directionally transmit RF energy for EH purposes. Notable research activities and major advances have occurred over the last decade in this direction. Thus, this paper provides a comprehensive survey of the state-of-art techniques, based on advances and open issues presented by simultaneous wireless information and power transfer (SWIPT) and WPT assisted technologies. More specifically, in contrast to the existing works, this paper identifies and provides a detailed description of various potential emerging technologies for the fifth generation communications with SWIPT/WPT. Moreover, we provide some interesting research challenges and recommendations with the objective of stimulating future research in this emerging domain.
621 citations
Cites methods from "How to Construct Polar Codes"
...A method for efficiently constructing polar codes has been presented and analysed in [180], and it has been applied in cooperative communications....
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TL;DR: In this paper, the secrecy capacity of the wiretap channel model was shown to be bounded by the mutual information between the message and the eavesdropper's observations, where the secrecy is defined as the probability of the receiver's probability of error in recovering the message.
Abstract: Suppose that Alice wishes to send messages to Bob through a communication channel C1, but her transmissions also reach an eavesdropper Eve through another channel C2. This is the wiretap channel model introduced by Wyner in 1975. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant Cs, called the secrecy capacity, which has the following meaning: for all e >; 0, there exist coding schemes of rate R ≥ Cs-e that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite consider able research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. In this paper, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C1 and C2 are symmetric and binary-input, and C2 is degraded with respect to C1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will also be satisfied unless the main channel C1 is noiseless, although we believe it can be always satisfied in practice.
455 citations
References
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TL;DR: The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I( W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n, rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate.
Abstract: A method is proposed, called channel polarization, to construct code sequences that achieve the symmetric capacity I(W) of any given binary-input discrete memoryless channel (B-DMC) W. The symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability. Channel polarization refers to the fact that it is possible to synthesize, out of N independent copies of a given B-DMC W, a second set of N binary-input channels {WN(i)1 les i les N} such that, as N becomes large, the fraction of indices i for which I(WN(i)) is near 1 approaches I(W) and the fraction for which I(WN(i)) is near 0 approaches 1-I(W). The polarized channels {WN(i)} are well-conditioned for channel coding: one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining. Codes constructed on the basis of this idea are called polar codes. The paper proves that, given any B-DMC W with I(W) > 0 and any target rate R< I(W) there exists a sequence of polar codes {Cfrn;nges1} such that Cfrn has block-length N=2n , rate ges R, and probability of block error under successive cancellation decoding bounded as Pe(N,R) les O(N-1/4) independently of the code rate. This performance is achievable by encoders and decoders with complexity O(N logN) for each.
3,554 citations
"How to Construct Polar Codes" refers methods in this paper
...Index Terms—channel polarization, channel degrading and upgrading, construction algorithms, polar codes I. INTRODUCTION POLAR codes, invented by Arıkan [3], achieve the capac-ity of arbitrary binary-input symmetric DMCs....
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...In this paper, however, we will restrict our attention to the original setting introduced by Arıkan in [3]....
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TL;DR: This summary of the state-of-the-art in iterative coding makes this decision more straightforward, with emphasis on the underlying theory, techniques to analyse and design practical iterative codes systems.
Abstract: Having trouble deciding which coding scheme to employ, how to design a new scheme, or how to improve an existing system? This summary of the state-of-the-art in iterative coding makes this decision more straightforward. With emphasis on the underlying theory, techniques to analyse and design practical iterative coding systems are presented. Using Gallager's original ensemble of LDPC codes, the basic concepts are extended for several general codes, including the practically important class of turbo codes. The simplicity of the binary erasure channel is exploited to develop analytical techniques and intuition, which are then applied to general channel models. A chapter on factor graphs helps to unify the important topics of information theory, coding and communication theory. Covering the most recent advances, this text is ideal for graduate students in electrical engineering and computer science, and practitioners. Additional resources, including instructor's solutions and figures, available online: www.cambridge.org/9780521852296.
2,100 citations
"How to Construct Polar Codes" refers background in this paper
...However, as indeed noted in [17], it is not clear how one would implement such convolutions to be sufficiently precise on one hand while being tractable on the other hand....
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03 Oct 2011
TL;DR: It appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes, and devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O( L · n) space.
Abstract: We describe a successive-cancellation list decoder for polar codes, which is a generalization of the classic successive-cancellation decoder of Arikan. In the proposed list decoder, up to L decoding paths are considered concurrently at each decoding stage. Simulation results show that the resulting performance is very close to that of a maximum-likelihood decoder, even for moderate values of L. Thus it appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes. The specific list-decoding algorithm that achieves this performance doubles the number of decoding paths at each decoding step, and then uses a pruning procedure to discard all but the L “best” paths. In order to implement this algorithm, we introduce a natural pruning criterion that can be easily evaluated. Nevertheless, straightforward implementation still requires O(L · n2) time, which is in stark contrast with the O(n log n) complexity of the original successive-cancellation decoder. We utilize the structure of polar codes to overcome this problem. Specifically, we devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O(L · n) space.
1,338 citations
"How to Construct Polar Codes" refers methods in this paper
...In [14], the polarization phenomenon has been studied for arbitrary kernel matrices, rather than Arıkan’s original 2× 2 polarization kernel, and error exponents were derived for each such kernel....
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28 Jun 2009
TL;DR: It is shown that, for any binary-input discrete memoryless channel W with symmetric capacity I(W) and any rate R ≪I(W), the polar-coding block-error probability under successive cancellation decoding satisfies Pe (N, R) ≤ 2−Nβ for any β ≪ 1/2 when the block-length N is large enough.
Abstract: A bound is given on the rate of channel polarization. As a corollary, an earlier bound on the probability of error for polar coding is improved. Specifically, it is shown that, for any binary-input discrete memoryless channel W with symmetric capacity I(W) and any rate R ≪ I(W), the polar-coding block-error probability under successive cancellation decoding satisfies P e (N, R) ≤ 2−Nβ for any β ≪ 1/2 when the block-length N is large enough.
486 citations
TL;DR: In this paper, the secrecy capacity of the wiretap channel model was shown to be bounded by the mutual information between the message and the eavesdropper's observations, where the secrecy is defined as the probability of the receiver's probability of error in recovering the message.
Abstract: Suppose that Alice wishes to send messages to Bob through a communication channel C1, but her transmissions also reach an eavesdropper Eve through another channel C2. This is the wiretap channel model introduced by Wyner in 1975. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant Cs, called the secrecy capacity, which has the following meaning: for all e >; 0, there exist coding schemes of rate R ≥ Cs-e that asymptotically achieve the reliability and security objectives. However, his proof of this result is based upon a random-coding argument. To date, despite consider able research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. In this paper, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C1 and C2 are symmetric and binary-input, and C2 is degraded with respect to C1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will also be satisfied unless the main channel C1 is noiseless, although we believe it can be always satisfied in practice.
455 citations